Will Deleting Complete Options Like Ships From Break Order Fulfillment?

Yes!  You cannot remove an entire option (like ships from, or color, or size, etc..) from a product in your shop.   You can however remove variations like the USA, green, and large if you don't want to offer those variations for example.   You must however leave one type of each option otherwise order fulfillment will break.

So for example if an item ships from USA and China, you can remove either USA or China, but cannot remove both.  
Also note that "Ships from" USA variants will only ship to USA addresses, same with the UK, Turkey, etc... "Ships from" China will ship internationally.

If you do not want your customers to see where an item is getting shipped from for example from China, what you can do is change Ships from China to Ships from "International Warehouse".  You can make those adjustments in the Edit Variations section before you Push an Item to your shop or by editing those variation names directly in your eCommerce platform (Shopify, Woocommerce...etc..)

Alternatively, if you remove all Shipping Countries but leave only one (like the USA for example), then the entire "Ships From" dropdown will not appear on your shop.  You can still fulfill orders for these products and even change the country of shipment by remapping variations.